How to combine proportional editing with vertex slide?
Proportional editing will alter the connected edges which isn't really compatible with using it as a reference to limit the transformation.

If you change the 3D manipulator to Normal you can restrict movement to an axis relative to the starting position of your geometry which might give the result you are looking for.

If the normal of the selected edges doesn't match what you are after you can make a custom orientation from the edge you want to follow before starting to move the vertices.